Next, it is easy to see that Lemma REF  holds and yields the same bound \(\Vert x_t-z_t\Vert \le \Vert \epsilon _t+\lambda _t^\top \delta _t\Vert /\sigma _{1:t}\)  with the redefined \(\lbrace \delta _t\rbrace \) . Applying [1]} to (REF ), we get:
\(&-\sum _{t=1}^T \lambda _t^\top g_t^\ell (z_t)+ \sum _{t=1}^T \lambda ^\top g_t^\ell (z_t)\\ &\le q_{0:T-1}(\lambda )+ \sum _{t=1}^T \Vert g_t^\ell (z_t)-\tilde{g}_t(\tilde{x}_t)\Vert _{(t-1),\star }^2= \\&=q_{0:T-1}(\lambda )+ \sum _{t=1}^T \frac{\Vert g_t^\ell (z_t)-\tilde{g}_t(\tilde{x}_t)\Vert ^2}{\phi _{0:t-1}},\,\,\,\forall \lambda \in \mathbb {R}_+^d. \) 
